Theorem negeu 4124

Description: Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18.

Hypotheses

Ref	Expression
negeu.1	|- A e. CC
negeu.2	|- B e. CC

Assertion

Ref	Expression
negeu	|- E!x e. CC (A + x) = B

Distinct variable group(s):   x,A   x,B

Proof of Theorem negeu

Step	Hyp	Ref	Expression
1		negeu.1	. . . . 5 |- A e. CC
2	1	negex 4116	. . . 4 |- E.y e. CC (A + y) = 0
3		negeu.2	. . . . . . . 8 |- B e. CC
4		axaddcl 4066	. . . . . . . 8 |- ((y e. CC /\ B e. CC) -> (y + B) e. CC)
5	3, 4	mpan2 519	. . . . . . 7 |- (y e. CC -> (y + B) e. CC)
6		risset 1235	. . . . . . 7 |- ((y + B) e. CC <-> E.x e. CC x = (y + B))
7	5, 6	sylib 173	. . . . . 6 |- (y e. CC -> E.x e. CC x = (y + B))
8		opreq2 3007	. . . . . . . . . . . . . . . 16 |- (x = (y + B) -> (A + x) = (A + (y + B)))
9		axaddass 4072	. . . . . . . . . . . . . . . . . . 19 |- ((A e. CC /\ y e. CC /\ B e. CC) -> ((A + y) + B) = (A + (y + B)))
10	1, 9	mp3an1 639	. . . . . . . . . . . . . . . . . 18 |- ((y e. CC /\ B e. CC) -> ((A + y) + B) = (A + (y + B)))
11	3, 10	mpan2 519	. . . . . . . . . . . . . . . . 17 |- (y e. CC -> ((A + y) + B) = (A + (y + B)))
12	11	cleqcomd 1106	. . . . . . . . . . . . . . . 16 |- (y e. CC -> (A + (y + B)) = ((A + y) + B))
13	8, 12	sylan9eqr 1145	. . . . . . . . . . . . . . 15 |- ((y e. CC /\ x = (y + B)) -> (A + x) = ((A + y) + B))
14		opreq1 3006	. . . . . . . . . . . . . . . 16 |- ((A + y) = 0 -> ((A + y) + B) = (0 + B))
15	3	addid2 4113	. . . . . . . . . . . . . . . 16 |- (0 + B) = B
16	14, 15	syl6eq 1140	. . . . . . . . . . . . . . 15 |- ((A + y) = 0 -> ((A + y) + B) = B)
17	13, 16	sylan9eqr 1145	. . . . . . . . . . . . . 14 |- (((A + y) = 0 /\ (y e. CC /\ x = (y + B))) -> (A + x) = B)
18	17	exp32 294	. . . . . . . . . . . . 13 |- ((A + y) = 0 -> (y e. CC -> (x = (y + B) -> (A + x) = B)))
19	18	com12 13	. . . . . . . . . . . 12 |- (y e. CC -> ((A + y) = 0 -> (x = (y + B) -> (A + x) = B)))
20	19	imp 277	. . . . . . . . . . 11 |- ((y e. CC /\ (A + y) = 0) -> (x = (y + B) -> (A + x) = B))
21	20	a1d 14	. . . . . . . . . 10 |- ((y e. CC /\ (A + y) = 0) -> (x e. CC -> (x = (y + B) -> (A + x) = B)))
22	21	r19.21aiv 1259	. . . . . . . . 9 |- ((y e. CC /\ (A + y) = 0) -> A.x e. CC (x = (y + B) -> (A + x) = B))
23	22	exp 291	. . . . . . . 8 |- (y e. CC -> ((A + y) = 0 -> A.x e. CC (x = (y + B) -> (A + x) = B)))
24		r19.22 1272	. . . . . . . 8 |- (A.x e. CC (x = (y + B) -> (A + x) = B) -> (E.x e. CC x = (y + B) -> E.x e. CC (A + x) = B))
25	23, 24	syl6 23	. . . . . . 7 |- (y e. CC -> ((A + y) = 0 -> (E.x e. CC x = (y + B) -> E.x e. CC (A + x) = B)))
26	25	com23 32	. . . . . 6 |- (y e. CC -> (E.x e. CC x = (y + B) -> ((A + y) = 0 -> E.x e. CC (A + x) = B)))
27	7, 26	mpd 46	. . . . 5 |- (y e. CC -> ((A + y) = 0 -> E.x e. CC (A + x) = B))
28	27	r19.23aiv 1284	. . . 4 |- (E.y e. CC (A + y) = 0 -> E.x e. CC (A + x) = B)
29	2, 28	ax-mp 6	. . 3 |- E.x e. CC (A + x) = B
30		addcant 4122	. . . . . 6 |- ((A e. CC /\ x e. CC /\ y e. CC) -> ((A + x) = (A + y) <-> x = y))
31		cleq2 1110	. . . . . . 7 |- ((A + y) = B -> ((A + x) = (A + y) <-> (A + x) = B))
32	31	biimparc 327	. . . . . 6 |- (((A + x) = B /\ (A + y) = B) -> (A + x) = (A + y))
33	30, 32	syl5bi 183	. . . . 5 |- ((A e. CC /\ x e. CC /\ y e. CC) -> (((A + x) = B /\ (A + y) = B) -> x = y))
34	1, 33	mp3an1 639	. . . 4 |- ((x e. CC /\ y e. CC) -> (((A + x) = B /\ (A + y) = B) -> x = y))
35	34	rgen2 1248	. . 3 |- A.x e. CC A.y e. CC (((A + x) = B /\ (A + y) = B) -> x = y)
36	29, 35	pm3.2i 234	. 2 |- (E.x e. CC (A + x) = B /\ A.x e. CC A.y e. CC (((A + x) = B /\ (A + y) = B) -> x = y))
37		opreq2 3007	. . . 4 |- (x = y -> (A + x) = (A + y))
38	37	cleq1d 1109	. . 3 |- (x = y -> ((A + x) = B <-> (A + y) = B))
39	38	reu4 1340	. 2 |- (E!x e. CC (A + x) = B <-> (E.x e. CC (A + x) = B /\ A.x e. CC A.y e. CC (((A + x) = B /\ (A + y) = B) -> x = y)))
40	36, 39	mpbir 165	1 |- E!x e. CC (A + x) = B

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  E!wreu 1203  (class class class)co 3001  CCcc 4026  0cc0 4028   + caddc 4031

This theorem is referenced by:  subcl 4139  subadd 4143

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-r 4038  df-plus 4039

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